The conventional or the stretch draw process to form a sheet into a part is generally divided into two stages: the binder-wrap stage and the die closure stage. The traditional method for sheet metal forming analysis is a quasi-static method. Due to the low speed of tool travel (e.g. less than 250 mm/S), the inertia effect of a thin sheet can be ignored. As a result, a quasi-static analysis can be justified and solved by the incremental method following the progress of tool movement.
A quasi-static analysis in three-dimensional space of the punch and die contact is described in the article titled "Analysis of Sheet Metal Stamping by a Finite-Element Method" authored by N. M. Wang and B. Budiansky, published by the Journal of Applied Mechanics, Vol. 45, No. 1, March 1978. The contribution of their method solved a contact problem in the quasi-static analysis by taking material derivatives of the contact forces. This, however, requires the curvature of tool surfaces, which is hard to obtain numerically. In addition, the coefficient matrix of the modified linear system governing the increment deformation of the sheet is no longer symmetric. The solution process was linearized by dividing the total tool travel distance into sufficiently small increments such that within a small interval the incremental equilibrium equation, which is approximately linear, could be solved. The method however, was based on the membrane shell theory, which has severe limitations, and the method cannot be applied to the analysis of a draw forming operation.
More recently, attempts have been made to analyze the punch and die contact utilizing crash worthiness programs based on an explicit time integration method. These programs are appropriate for structures dominated by an inertia load, such as the transient response to an automobile crash. As a result, the explicit time integration method yields inaccurate results for a sheet forming analysis without artificial adjustments for forming speeds and damping parameters which are problem dependent.
Due to severe discontinuities and the occurrence of structural instability, traditional numerical solutions often have convergence problems. A method and apparatus for addressing and resolving these convergence problems utilizing a more reliable implicit time integration method, so as to avoid undesirable oscillation in the transient response during forming, was developed and is described in issued U.S. patent application Ser. No. 07/994,014, filed Dec. 21, 1992, issued Sep. 1, 1994 (Pat. No. unassigned), titled Method For Aiding Sheet Metal Forming Tooling Design, assigned to the assignee of the present invention, the specification of which is hereby expressly incorporated by reference in its entirety.
In both the binder and die closure analysis, snap-through-type buckling may occur. Close to the onset of buckling, the tangent stiffness matrix in the Newton-Raphson method is ill-conditioned or singular because it reflects the stiffness of an unstable structure. This causes the iterative solution process to diverge because of the large nodal displacement increments that are computed based upon this matrix. Existing methods, such as the constant arc method, are insufficient since they do not address the contact problem. Accordingly, it would be desirable to address and resolve the divergence problems.